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# Exceptional Sets of Slices for Functions From the Bergman Space in the Ball

Published:2001-06-01
Printed: Jun 2001
• Piotr Jakóbczak
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## Abstract

Let $B_N$ be the unit ball in $\mathbb{C}^N$ and let $f$ be a function holomorphic and $L^2$-integrable in $B_N$. Denote by $E(B_N,f)$ the set of all slices of the form $\Pi =L\cap B_N$, where $L$ is a complex one-dimensional subspace of $\mathbb{C}^N$, for which $f|_{\Pi}$ is not $L^2$-integrable (with respect to the Lebesgue measure on $L$). Call this set the exceptional set for $f$. We give a characterization of exceptional sets which are closed in the natural topology of slices.
 MSC Classifications: 32A37 - Other spaces of holomorphic functions (e.g. bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) [See also 46Exx] 32A22 - Nevanlinna theory (local); growth estimates; other inequalities {For geometric theory, see 32H25, 32H30}